Heat your oven to 350 degrees F and line two baking sheets with parchment paper. Grate the zucchini on the large holes of a box grater. Transfer to a clean kitchen towel and squeeze it firmly over the sink, wringing out as much liquid as you can. You need it very dry or the cookies will spread too much. Set the dry zucchini shreds aside.

Whisk together the flour, baking soda, baking powder, salt, and cinnamon in a medium bowl. The cinnamon is subtle but it rounds out the butterscotch flavor in a nice way.

In a large bowl, beat the softened butter, brown sugar, and granulated sugar on medium speed for about 2 minutes until light and creamy. Add the eggs one at a time, beating well after each. Mix in the vanilla extract.

Add the flour mixture to the butter mixture and stir with a rubber spatula until just combined. Fold in the squeezed zucchini and then the butterscotch chips. The dough will be soft and a little sticky, which is normal for a zucchini cookie.

Scoop the dough using a 1 1/2 tablespoon cookie scoop, placing the mounds about 2 inches apart on the lined baking sheets. Bake one sheet at a time on the center oven rack for 11 to 13 minutes. You are looking for edges that are just golden and set. The centers may look soft, and that is exactly what you want. Pull them when the edges color up, even if the middle still looks a tiny bit underdone.

Let the cookies cool on the baking sheet for 5 minutes before moving them to a wire rack. They need that rest time to firm up enough to lift without breaking. Cool completely before storing.